Question: Simplify the following expression: $y = \dfrac{8x^2+31x- 4}{8x - 1}$
Solution: First use factoring by grouping to factor the expression in the numerator. This expression is in the form ${A}x^2 + {B}x + {C}$ First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(8)}{(-4)} &=& -32 \\ {a} + {b} &=& &=& {31} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-32$ and add them together. Remember, since $-32$ is negative, one of the factors must be negative. The factors that add up to ${31}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-1}$ and ${b}$ is ${32}$ $ \begin{eqnarray} {ab} &=& ({-1})({32}) &=& -32 \\ {a} + {b} &=& {-1} + {32} &=& 31 \end{eqnarray} $ Next, rewrite the expression as $({A}x^2 + {a}x) + ({b}x + {C})$ $ ({8}x^2 {-1}x) + ({32}x {-4}) $ Factor out the common factors: $ x(8x - 1) + 4(8x - 1)$ Now factor out $(8x - 1)$ $ (8x - 1)(x + 4)$ The original expression can therefore be written: $ \dfrac{(8x - 1)(x + 4)}{8x - 1}$ We are dividing by $8x - 1$ , so $8x - 1 \neq 0$ Therefore, $x \neq \frac{1}{8}$ This leaves us with $x + 4; x \neq \frac{1}{8}$.